Access
You are not currently logged in.
Access your personal account or get JSTOR access through your library or other institution:
If You Use a Screen Reader
This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.Asymptotic Distribution of the Likelihood Function in the Independent not Identically Distributed Case
A. N. Philippou and G. G. Roussas
The Annals of Statistics
Vol. 1, No. 3 (May, 1973), pp. 454471
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2958104
Page Count: 18
 Item Type
 Article
 Thumbnails
 References
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Abstract
Let Θ be an open subset of Rk and for each θ ∈ Θ, let X1, ⋯, Xn be independent rv's defined on the probability space (X, A, Pθ), and let pj, θ be the distribution of the rv Xj. Let $f_j(\bullet; \theta)$ be a specified version of the RadonNikodym derivative of pj, θ with respect to a σfinite measure μ and set fj(θ) = fj(Xj; θ). Furthermore, for θ, θ* ∈ Θ, set φj(θ, θ*) = [ fj(θ*)/fj(θ)]1/2 and suppose that φj(θ, θ*) is differentiable in quadratic mean (qm) with respect to θ* at (θ, θ), when the probability measure Pθ is employed, with qm derivative φ̇j(θ). Set Δn(θ) = 2n1/2 ∑n j=1 φ̇j(θ), Γj(θ) = 4Eθ[φ̇j(θ)φ̇j'(θ)], Γ̄n(θ) = n1 ∑n j=1 Γj(θ), and suppose that Γ̄n(θ) → Γ̄(θ) and Γ̄(θ) is positive definite on Θ. Finally, for hn → h ∈ Rk, set θn = θ + hnn 1/2 and Λn(θ) = log[ dPn,θn /dPn,θ], where Pn,θ stands for the restriction of Pθ to An = σ(X1, ⋯, Xn). Then, under suitableand not too hard to verifyconditions, we obtain, the following results. The limits are taken as n → ∞. THEOREM 1. Λn(θ)  h'Δn(θ) → 1/2 h'Γ̄(θ)h in Pθprobability, θ ∈ Θ. THEOREM 2. $\mathscr{L}\lbrack\Delta_n(\theta) \mid P_\theta\rbrack \Rightarrow N(0, \bar{\Gamma}(\theta)), \theta \in \Theta$. THEOREM 3. $\mathscr{L}\lbrack\Lambda_n(\theta) \mid P_\theta\rbrack \Rightarrow N(\frac{1}{2} h'\bar{\Gamma}(\theta)h, h'\bar{\Gamma}(\theta)h), \theta \in \Theta$. THEOREM 4. Λn(θ)  h'Δn(θ) → 1/2h'Γ̄(θ)h in Pθn probability, θ ∈ Θ. THEOREM 5. $\mathscr{L}\lbrack\Lambda_n(\theta) \mid P_{\theta_n}\rbrack \Rightarrow N(\frac{1}{2}h'\bar{\Gamma}(\theta)h, h'\bar{\Gamma}(theta)h), \theta \in \Theta$. THEOREM 6. $\mathscr{L}\lbrack\Delta_n(\theta) \mid P_{\theta_n}\rbrack \Rightarrow N(\bar{\Gamma}(\theta)h, \bar{\Gamma}(\theta)), \theta \in \Theta$.
Page Thumbnails

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471
The Annals of Statistics © 1973 Institute of Mathematical Statistics